Representation of uncertainty by interval valued fuzzy setsapplication to image thresholding

Resumen

In this dissertation we study image thresholding by means of fuzzy techniques, The more important advantage of a fuzzy methodology lies in that the fuzzy membership function provides a natural means to model the uncertainty in an image. A key problem during the design of fuzzy systems is the election of themembership functions that are going to represent the variables of the system, in this case the membership function that best represents the image. To deal with this problem arise the main objective of the dissertation:To incorporate within the thresholding algorithm the uncertainty (hesitation) that the expert has in the election of the best membership function, in order to obtain better thresholds.In chapter 1 we present a brief introduction about thresholding algorithms, devoting special interest to classical early thresholding algorithms and first fuzzy algorithms. We explain their foundations and the problems arising. Furthermore, we do an introduction of the basic concepts of the Interval valued fuzzy sets theory that we will use throughout the dissertation.In chapter 2 we go deeply in the classical fuzzy thresholding algorithm, particularly in its drawbacks. We propose a generalization of the IVFS algorithm proposed by Tizhoosh and we prove that we must always consider the minimum entropy.In chapter 3 we develop a new algorithm. To do so, we enforce that the expert select several membership functions. From these membership functions, we show an IVFSs construction method in such a way that the lengths of the membership intervals represent the uncertainty of the expert when assigning to each pixel a specific membership value.In chapter 4 we study a modification of the classical fuzzy algorithm. We force the user to choose a membership function to represent the background and another to represent the object. From these functions we define the unknowledge functions, which allow us to estimate the uncertainty the expert has had in the election of such functions.In chapter 5 we study the way to recover the fuzzy algorithm from any of the IVFS algorithm presented in chapters 2, 3 and 4, using operator Krho. We study the necessary conditions that should be demanded to the fuzzy measures devoted to image comparison. We prove that the measure most used to compare images is a particular case of the measures that we propose.